Add generated VHDL to the hardware description chapter.

[matthijs/master-project/report.git] / Chapters / HardwareDescription.tex

diff --git a/Chapters/HardwareDescription.tex b/Chapters/HardwareDescription.tex
index 9e4061c9a4d059ee7fd8182c9d1e04565ecf94df..479dd400640b5dfa38de5a2c929d41ef79f75a54 100644 (file)
--- a/Chapters/HardwareDescription.tex
+++ b/Chapters/HardwareDescription.tex
@@ -1,38 +1,78 @@
 \chapter[chap:description]{Hardware description}
 \chapter[chap:description]{Hardware description}

-  This chapter will provide an overview of the hardware description language
-  that was created and the issues that have arisen in the process. It will
-  focus on the issues of the language, not the implementation.
-
-  When translating Haskell to hardware, we need to make choices in what kind
-  of hardware to generate for what Haskell constructs. When faced with
-  choices, we've tried to stick with the most obvious choice wherever
-  possible. In a lot of cases, when you look at a hardware description it is
-  comletely clear what hardware is described. We want our translator to
-  generate exactly that hardware whenever possible, to minimize the amount of
-  surprise for people working with it.
+  In this chapter an overview will be provided of the hardware
+  description language that was created and the issues that have arisen
+  in the process. The focus will be on the issues of the language, not
+  the implementation. The prototype implementation will be discussed in
+  \in{chapter}[chap:prototype].
+
+  To translate Haskell to hardware, every Haskell construct needs a
+  translation to \VHDL. There are often multiple valid translations
+  possible. When faced with choices, the most obvious choice has been
+  chosen wherever possible. In a lot of cases, when a programmer looks
+  at a functional hardware description it is completely clear what
+  hardware is described. We want our translator to generate exactly that
+  hardware whenever possible, to make working with Cλash as intuitive as
+  possible.

    
    

-  In this chapter we try to describe how we interpret a Haskell program from a
+  \placeintermezzo{}{
+    \defref{reading examples}
+    \startframedtext[width=9cm,background=box,frame=no]
+    \startalignment[center]
+      {\tfa Reading the examples}
+    \stopalignment
+    \blank[medium]
+      In this thesis, a lot of functional code will be presented. Part of this
+      will be valid Cλash code, but others will just be small Haskell or Core
+      snippets to illustrate a concept.
+
+      In these examples, some functions and types will be used, without
+      properly defining every one of them. These functions (like \hs{and},
+      \hs{not}, \hs{add}, \hs{+}, etc.) and types (like \hs{Bit}, \hs{Word},
+      \hs{Bool}, etc.) are usually pretty self-explanatory.
+
+      The special type \hs{[t]} means \quote{list of \hs{t}'s}, where \hs{t}
+      can be any other type.
+
+      Of particular note is the use of the \hs{::} operator. It is used in
+      Haskell to explicitly declare the type of function or let binding. In
+      these examples and the text, we will occasionally use this operator to
+      show the type of arbitrary expressions, even where this would not
+      normally be valid. Just reading the \hs{::} operator as \quote{and also
+      note that \emph{this} expression has \emph{this} type} should work out.
+    \stopframedtext
+  }
+
+  In this chapter we describe how to interpret a Haskell program from a

   hardware perspective. We provide a description of each Haskell language
   element that needs translation, to provide a clear picture of what is
   supported and how.
 
   hardware perspective. We provide a description of each Haskell language
   element that needs translation, to provide a clear picture of what is
   supported and how.
 

-  \section{Function application}
-  The basic syntactic element of a functional program are functions and
-  function application. These have a single obvious \small{VHDL} translation: Each
-  function becomes a hardware component, where each argument is an input port
-  and the result value is the output port.
+
+  \section[sec:description:application]{Function application}
+  The basic syntactic elements of a functional program are functions and
+  function application. These have a single obvious \small{VHDL}
+  translation: each top level function becomes a hardware component, where each
+  argument is an input port and the result value is the (single) output
+  port. This output port can have a complex type (such as a tuple), so
+  having just a single output port does not pose a limitation.

 
   Each function application in turn becomes component instantiation. Here, the
   result of each argument expression is assigned to a signal, which is mapped
   to the corresponding input port. The output port of the function is also
   mapped to a signal, which is used as the result of the application.
 
 
   Each function application in turn becomes component instantiation. Here, the
   result of each argument expression is assigned to a signal, which is mapped
   to the corresponding input port. The output port of the function is also
   mapped to a signal, which is used as the result of the application.
 

+  Since every top level function generates its own component, the
+  hierarchy of of function calls is reflected in the final \VHDL\ output
+  as well, creating a hierarchical \VHDL\ description of the hardware.
+  This separation in different components makes the resulting \VHDL\
+  output easier to read and debug.
+

   \in{Example}[ex:And3] shows a simple program using only function
   application and the corresponding architecture.
 
 \startbuffer[And3]
   \in{Example}[ex:And3] shows a simple program using only function
   application and the corresponding architecture.
 
 \startbuffer[And3]

--- | A simple function that returns 
---   the and of three bits
+-- A simple function that returns 
+-- conjunction of three bits

 and3 :: Bit -> Bit -> Bit -> Bit
 and3 a b c = and (and a b) c
 \stopbuffer
 and3 :: Bit -> Bit -> Bit -> Bit
 and3 a b c = and (and a b) c
 \stopbuffer


@@ -68,49 +108,147 @@ and3 a b c = and (and a b) c
     ncline(andb)(out);
   \stopuseMPgraphic
 
     ncline(andb)(out);
   \stopuseMPgraphic
 

-  \placeexample[here][ex:And3]{Simple three port and.}
+  \startbuffer[And3VHDL]
+    entity and3Component_0 is
+         port (\azMyG2\ : in std_logic;
+               \bzMyI2\ : in std_logic;
+               \czMyK2\ : in std_logic;
+               \foozMySzMyS2\ : out std_logic;
+               clock : in std_logic;
+               resetn : in std_logic);
+    end entity and3Component_0;
+
+
+    architecture structural of and3Component_0 is
+         signal \argzMyMzMyM2\ : std_logic;
+    begin
+         \argzMyMzMyM2\ <= \azMyG2\ and \bzMyI2\;
+
+         \foozMySzMyS2\ <= \argzMyMzMyM2\ and \czMyK2\;
+    end architecture structural;
+  \stopbuffer
+
+  \placeexample[][ex:And3]{Simple three input and gate.}

     \startcombination[2*1]
       {\typebufferhs{And3}}{Haskell description using function applications.}
       {\boxedgraphic{And3}}{The architecture described by the Haskell description.}
     \stopcombination
 
     \startcombination[2*1]
       {\typebufferhs{And3}}{Haskell description using function applications.}
       {\boxedgraphic{And3}}{The architecture described by the Haskell description.}
     \stopcombination
 

-  TODO: Define top level function and subfunctions/circuits.
+  \placeexample[][ex:And3VHDL]{\VHDL\ generated for \hs{and3} from \in{example}[ex:And3]}
+      {\typebuffervhdl{And3VHDL}}
+
+  \placeintermezzo{}{
+    \defref{top level binder}
+    \defref{top level function}
+    \startframedtext[width=8cm,background=box,frame=no]
+    \startalignment[center]
+      {\tfa Top level binders and functions}
+    \stopalignment
+    \blank[medium]
+      A top level binder is any binder (variable) that is declared in
+      the \quote{global} scope of a Haskell program (as opposed to a
+      binder that is bound inside a function.
+
+      In Haskell, there is no sharp distinction between a variable and a
+      function: a function is just a variable (binder) with a function
+      type. This means that a top level function is just any top level
+      binder with a function type.
+
+      As an example, consider the following Haskell snippet:

 
 

+      \starthaskell
+        foo :: Int -> Int
+        foo x = inc x
+          where
+            inc = \y -> y + 1
+      \stophaskell
+
+      Here, \hs{foo} is a top level binder, whereas \hs{inc} is a
+      function (since it is bound to a lambda extraction, indicated by
+      the backslash) but is not a top level binder or function.  Since
+      the type of \hs{foo} is a function type, namely \hs{Int -> Int},
+      it is also a top level function.
+    \stopframedtext
+  }

   \section{Choice}
   \section{Choice}

-    Since describing components and connections allows us to describe a lot of
-    hardware designs already, there is an obvious thing missing: Choice. We
+    Although describing components and connections allows us to describe a lot of
+    hardware designs already, there is an obvious thing missing: choice. We

     need some way to be able to choose between values based on another value.
     need some way to be able to choose between values based on another value.

-    In Haskell, choice is achieved by case expressions, if expressions or
-    pattern matching (all of which can be reduced to case expressions).
+    In Haskell, choice is achieved by \hs{case} expressions, \hs{if}
+    expressions, pattern matching and guards.

 
     An obvious way to add choice to our language without having to recognize
     any of Haskell's syntax, would be to add a primivite \quote{\hs{if}}
 
     An obvious way to add choice to our language without having to recognize
     any of Haskell's syntax, would be to add a primivite \quote{\hs{if}}

-    function. This function would take three arguments: The condition, the
+    function. This function would take three arguments: the condition, the

     value to return when the condition is true and the value to return when
     the condition is false.
 
     This \hs{if} function would then essentially describe a multiplexer and
     value to return when the condition is true and the value to return when
     the condition is false.
 
     This \hs{if} function would then essentially describe a multiplexer and

-    allows us to describe any architecture that uses multiplexers. (TODO: Are
-    there other mechanisms of choice in hardware?)
+    allows us to describe any architecture that uses multiplexers.

 
     However, to be able to describe our hardware in a more convenient way, we
 
     However, to be able to describe our hardware in a more convenient way, we

-    also need to translate Haskell's choice mechanisms. The easiest of these
-    are of course case expressions (and if expressions, which can be very
-    directly translated to case expressions). A case expression can in turn
+    also want to translate Haskell's choice mechanisms. The easiest of these
+    are of course case expressions (and \hs{if} expressions, which can be very
+    directly translated to \hs{case} expressions). A \hs{case} expression can in turn

     simply be translated to a conditional assignment, where the conditions use
     simply be translated to a conditional assignment, where the conditions use

-    equality comparisons against the constructors in the case expressions.
+    equality comparisons against the constructors in the \hs{case} expressions.
+
+    In \in{example}[ex:CaseInv] a simple \hs{case} expression is shown,
+    scrutinizing a boolean value. The corresponding architecture has a
+    comparator to determine which of the constructors is on the \hs{in}
+    input. There is a multiplexer to select the output signal. The two options
+    for the output signals are just constants, but these could have been more
+    complex expressions (in which case also both of them would be working in
+    parallel, regardless of which output would be chosen eventually).
+
+    If we would translate a Boolean to a bit value, we could of course remove
+    the comparator and directly feed 'in' into the multiplexer (or even use an
+    inverter instead of a multiplexer).  However, we will try to make a
+    general translation, which works for all possible \hs{case} expressions.
+    Optimizations such as these are left for the \VHDL\ synthesizer, which
+    handles them very well.
+
+    \placeintermezzo{}{
+      \startframedtext[width=8cm,background=box,frame=no]
+      \startalignment[center]
+        {\tfa Arguments / results vs. inputs / outputs}
+      \stopalignment
+      \blank[medium]
+        Due to the translation chosen for function application, there is a
+        very strong relation between arguments, results, inputs and outputs.
+        For clarity, the former two will always refer to the arguments and
+        results in the functional description (either Haskell or Core). The
+        latter two will refer to input and output ports in the generated
+        \VHDL.
+
+        Even though these concepts seem to be nearly identical, when stateful
+        functions are introduces we will see arguments and results that will
+        not get translated into input and output ports, making this
+        distinction more important.
+      \stopframedtext
+    }
+
+    A slightly more complex (but very powerful) form of choice is pattern
+    matching. A function can be defined in multiple clauses, where each clause
+    specifies a pattern. When the arguments match the pattern, the
+    corresponding clause will be used.
+
+    The architecture described by \in{example}[ex:PatternInv] is of course the
+    same one as the one in \in{example}[ex:CaseInv]. The general interpretation
+    of pattern matching is also similar to that of \hs{case} expressions: generate
+    hardware for each of the clauses (like each of the clauses of a \hs{case}
+    expression) and connect them to the function output through (a number of
+    nested) multiplexers. These multiplexers are driven by comparators and
+    other logic, that check each pattern in turn.

 
 

-    In \in{example}[ex:CaseInv] a simple case statement is shown,
-    scrutinizing a boolean value. If we would translate a Boolean to a bit
-    value, we could of course remove the comparator and directly feed 'in'
-    into the multiplex (or even use an inverter instead of a multiplexer).
-    However, we will try to make a general translation, which works for all
-    possible case statements. Optimizations such as these are left for the
-    \VHDL synthesizer, which handles them very well.
+    In these examples we have seen only binary case expressions and pattern
+    matches (\ie, with two alternatives). In practice, case expressions can
+    choose between more than two values, resulting in a number of nested
+    multiplexers.

 
     \startbuffer[CaseInv]
     inv :: Bool -> Bool
 
     \startbuffer[CaseInv]
     inv :: Bool -> Bool

-    inv in = case in of
+    inv x = case x of

       True -> False
       False -> True
     \stopbuffer
       True -> False
       False -> True
     \stopbuffer


@@ -148,17 +286,32 @@ and3 a b c = and (and a b) c
       ncline(trueout)(mux) "posB(inpb)";
       ncline(mux)(out) "posA(out)";
     \stopuseMPgraphic
       ncline(trueout)(mux) "posB(inpb)";
       ncline(mux)(out) "posA(out)";
     \stopuseMPgraphic

+    
+    \startbuffer[CaseInvVHDL]
+      entity invComponent_0 is
+           port (\xzAMo2\ : in boolean;
+                 \reszAMuzAMu2\ : out boolean;
+                 clock : in std_logic;
+                 resetn : in std_logic);
+      end entity invComponent_0;
+
+
+      architecture structural of invComponent_0 is
+      begin
+           \reszAMuzAMu2\ <= false when \xzAMo2\ = true else
+                             true;
+      end architecture structural; 
+    \stopbuffer

 
 

-    \placeexample[here][ex:CaseInv]{Simple inverter.}
+    \placeexample[][ex:CaseInv]{Simple inverter.}

       \startcombination[2*1]
         {\typebufferhs{CaseInv}}{Haskell description using a Case expression.}
         {\boxedgraphic{CaseInv}}{The architecture described by the Haskell description.}
       \stopcombination
 
       \startcombination[2*1]
         {\typebufferhs{CaseInv}}{Haskell description using a Case expression.}
         {\boxedgraphic{CaseInv}}{The architecture described by the Haskell description.}
       \stopcombination
 

-    Slightly more complex (but very powerful) forms of choice are pattern
-    matches. A function can be defined in multiple clauses, where each clause
-    specifies a pattern. When the arguments match the pattern, the
-    corresponding clause will be used.
+    \placeexample[][ex:CaseInvVHDL]{\VHDL\ generated for \hs{inv} from
+    \in{example}[ex:CaseInv] and \in{example}[ex:PatternInv]}
+        {\typebuffervhdl{CaseInvVHDL}}

 
     \startbuffer[PatternInv]
     inv :: Bool -> Bool
 
     \startbuffer[PatternInv]
     inv :: Bool -> Bool


@@ -166,50 +319,60 @@ and3 a b c = and (and a b) c
     inv False = True
     \stopbuffer
 
     inv False = True
     \stopbuffer
 

-    \placeexample[here][ex:PatternInv]{Simple inverter using pattern matching.}
-        {\typebufferhs{CaseInv}}
-
-    The architecture described in \in{example}[ex:PatternInv] is of course the
-    same one as the one in \in{example}[ex:CaseInv]. The general interpration
-    of pattern matching is also similar to that of case expressions: Generate
-    hardware for each of the clause (like each of the clauses of a case
-    expression) and connect them to the function output through (a number of
-    nested) multiplexers. These multiplexers are driven by comparators and
-    other logic, that check each pattern in turn.
+    \placeexample[][ex:PatternInv]{Simple inverter using pattern matching.
+    Describes the same architecture as \in{example}[ex:CaseInv].}
+        {\typebufferhs{PatternInv}}

 
   \section{Types}
 
   \section{Types}

-    Apart from giving structure to our hardware, we'll also need to provide
-    some way to translate the values used to hardware equivalents. In
-    particular, this means having to come up with a hardware equivalent for
-    every \emph{type} used in our program.
-
-    Since most functional languages have a lot of standard types that are
-    hard to translate (integers without a fixed size, lists without a static
-    length, etc.), we will start out by defining a number of \quote{builtin}
-    types ourselves. These types are builtin in the sense that our compiler
-    will have a fixed VHDL type for these. User defined types, on the other
-    hand, will have their hardware type derived directly from their Haskell
-    declaration automatically, according to the rules we sketch here.
-
-    \subsection{Builtin types}
-      \startdesc{Bit}
+    Translation of two most basic functional concepts has been
+    discussed: function application and choice. Before looking further
+    into less obvious concepts like higher-order expressions and
+    polymorphism, the possible types that can be used in hardware
+    descriptions will be discussed.
+
+    Some way is needed to translate every values used to its hardware
+    equivalents. In particular, this means a hardware equivalent for
+    every \emph{type} used in a hardware description is needed
+
+    Since most functional languages have a lot of standard types that
+    are hard to translate (integers without a fixed size, lists without
+    a static length, etc.), a number of \quote{built-in} types will be
+    defined first. These types are built-in in the sense that our
+    compiler will have a fixed VHDL type for these. User defined types,
+    on the other hand, will have their hardware type derived directly
+    from their Haskell declaration automatically, according to the rules
+    sketched here.
+
+    \todo{Introduce Haskell type syntax (type constructors, type application,
+    :: operator?)}
+
+    \subsection{Built-in types}
+      The language currently supports the following built-in types. Of these,
+      only the \hs{Bool} type is supported by Haskell out of the box (the
+      others are defined by the Cλash package, so they are user-defined types
+      from Haskell's point of view).
+
+      \startdesc{\hs{Bit}}

         This is the most basic type available. It is mapped directly onto
         the \type{std_logic} \small{VHDL} type. Mapping this to the
         \type{bit} type might make more sense (since the Haskell version
         only has two values), but using \type{std_logic} is more standard
         (and allowed for some experimentation with don't care values)
         This is the most basic type available. It is mapped directly onto
         the \type{std_logic} \small{VHDL} type. Mapping this to the
         \type{bit} type might make more sense (since the Haskell version
         only has two values), but using \type{std_logic} is more standard
         (and allowed for some experimentation with don't care values)

+
+        \todo{Sidenote bit vs stdlogic}

       \stopdesc
       \stopdesc

-      \startdesc{Bool}
-        This is a builtin Haskell type, but is translated exactly like the
-        Bit type. Supporting the Bool type is particularly useful to support
-        \hs{if ... then ... else ...} expressions, which always have a
-        \hs{Bool} value for the condition.
+      \startdesc{\hs{Bool}}
+        This is the only built-in Haskell type supported and is translated
+        exactly like the Bit type (where a value of \hs{True} corresponds to a
+        value of \hs{High}). Supporting the Bool type is particularly
+        useful to support \hs{if ... then ... else ...} expressions, which
+        always have a \hs{Bool} value for the condition.

 
         A \hs{Bool} is translated to a \type{std_logic}, just like \hs{Bit}.
       \stopdesc
 
         A \hs{Bool} is translated to a \type{std_logic}, just like \hs{Bit}.
       \stopdesc

-      \startdesc{SizedWord, SizedInt}
-        These are types to represent integers. \hs{SizedWord} is unsigned,
-        while \hs{SizedInt} is signed. These types are parameterized by a
+      \startdesc{\hs{SizedWord}, \hs{SizedInt}}
+        These are types to represent integers. A \hs{SizedWord} is unsigned,
+        while a \hs{SizedInt} is signed. These types are parameterized by a

         length type, so you can define an unsigned word of 32 bits wide as
         follows:
         
         length type, so you can define an unsigned word of 32 bits wide as
         follows:
         


@@ -217,67 +380,104 @@ and3 a b c = and (and a b) c
           type Word32 = SizedWord D32
         \stophaskell
 
           type Word32 = SizedWord D32
         \stophaskell
 

-        Here, \hs{D32} is the \emph{type level representation} of the number
-        32. TODO: Introduce dependent types somewhere.
+        Here, a type synonym \hs{Word32} is defined that is equal to the
+        \hs{SizedWord} type constructor applied to the type \hs{D32}. \hs{D32}
+        is the \emph{type level representation} of the decimal number 32,
+        making the \hs{Word32} type a 32-bit unsigned word.

 
         These types are translated to the \small{VHDL} \type{unsigned} and
         \type{signed} respectively.
 
         These types are translated to the \small{VHDL} \type{unsigned} and
         \type{signed} respectively.

+        \todo{Sidenote on dependent typing?}

       \stopdesc
       \stopdesc

-      \startdesc{RangedWord}
+      \startdesc{\hs{Vector}}
+        This is a vector type, that can contain elements of any other type and
+        has a fixed length. It has two type parameters: its
+        length and the type of the elements contained in it. By putting the
+        length parameter in the type, the length of a vector can be determined
+        at compile time, instead of only at runtime for conventional lists.
+
+        The \hs{Vector} type constructor takes two type arguments: the length
+        of the vector and the type of the elements contained in it. The state
+        type of an 8 element register bank would then for example be:
+
+        \starthaskell
+        type RegisterState = Vector D8 Word32
+        \stophaskell
+
+        Here, a type synonym \hs{RegisterState} is defined that is equal to
+        the \hs{Vector} type constructor applied to the types \hs{D8} (The type
+        level representation of the decimal number 8) and \hs{Word32} (The 32
+        bit word type as defined above). In other words, the
+        \hs{RegisterState} type is a vector of 8 32-bit words.
+
+        A fixed size vector is translated to a \small{VHDL} array type.
+      \stopdesc
+      \startdesc{\hs{RangedWord}}

         This is another type to describe integers, but unlike the previous
         This is another type to describe integers, but unlike the previous

-        two it has no specific width, but an upper bound. This means that
+        two it has no specific bitwidth, but an upper bound. This means that

         its range is not limited to powers of two, but can be any number.
         A \hs{RangedWord} only has an upper bound, its lower bound is
         implicitly zero. There is a lot of added implementation complexity
         when adding a lower bound and having just an upper bound was enough
         its range is not limited to powers of two, but can be any number.
         A \hs{RangedWord} only has an upper bound, its lower bound is
         implicitly zero. There is a lot of added implementation complexity
         when adding a lower bound and having just an upper bound was enough

-        for the primary purpose of this type: Typesafely indexing vectors.
+        for the primary purpose of this type: typesafely indexing vectors.
+
+        To define an index for the 8 element vector above, we would do:
+
+        \starthaskell
+        type RegisterIndex = RangedWord D7
+        \stophaskell
+
+        Here, a type synonym \hs{RegisterIndex} is defined that is equal to
+        the \hs{RangedWord} type constructor applied to the type \hs{D7}. In
+        other words, this defines an unsigned word with values from
+        {\definedfont[Serif*normalnum]0 to 7} (inclusive). This word can be be used to index the
+        8 element vector \hs{RegisterState} above.

 
         This type is translated to the \type{unsigned} \small{VHDL} type.
       \stopdesc
 
         This type is translated to the \type{unsigned} \small{VHDL} type.
       \stopdesc

-      \startdesc{TFVec}
-        This is a vector type, with a fixed length. It has two type
-        parameters: Its length and the type of the elements contained in it.

 
 

-        A fixed size vector is translated to a \small{VHDL} array type.
-       
-        The \hs{TF} in its name is a reference to the implementation used in
-        the prototype, which uses \emph{type families}.
-      \stopdesc
+      The integer and vector built-in types are discussed in more detail
+      in \cite[baaij09].

 
 

-      TODO: Reference Christiaan / describe dependent typing

     \subsection{User-defined types}
     \subsection{User-defined types}

-      There are three ways to define new types in Haskell: Algebraic
+      There are three ways to define new types in Haskell: algebraic

       datatypes with the \hs{data} keyword, type synonyms with the \hs{type}
       datatypes with the \hs{data} keyword, type synonyms with the \hs{type}

-      keyword and type renamings with the \hs{newtype} keyword. This
-      expclitely excludes more advanced type creation from \GHC extensions
-      such as type families, existential typing, \small{GADT}s, etc.
-
-      The first of these actually introduces a new type, for which we provide
-      the \VHDL translation below. The latter two only define new names for
+      keyword and type renamings with the \hs{newtype} keyword. \GHC\
+      offers a few more advanced ways to introduce types (type families,
+      existential typing, \small{GADT}s, etc.) which are not standard
+      Haskell.  These will be left outside the scope of this research.
+
+      Only an algebraic datatype declaration actually introduces a
+      completely new type, for which we provide the \VHDL\ translation
+      below. Type synonyms and renamings only define new names for

       existing types (where synonyms are completely interchangeable and
       existing types (where synonyms are completely interchangeable and

-      renamings need explicit conversion). Therefore, these don't need any
-      particular \VHDL translation, a synonym or renamed type will just use
-      the same representation as the equivalent type.
+      renamings need explicit conversion). Therefore, these do not need
+      any particular \VHDL\ translation, a synonym or renamed type will
+      just use the same representation as the original type. The
+      distinction between a renaming and a synonym does no longer matter
+      in hardware and can be disregarded in the generated \VHDL.

 
       For algebraic types, we can make the following distinction:
 
       \startdesc{Product types}
         A product type is an algebraic datatype with a single constructor with
 
       For algebraic types, we can make the following distinction:
 
       \startdesc{Product types}
         A product type is an algebraic datatype with a single constructor with

-        two or more fields. This is essentially a way to pack a few values
-        together in a record-like structure. In fact, the builtin tuple types
-        are just product types (and are thus supported in exactly the same
-        way).
+        two or more fields, denoted in practice like (a,b), (a,b,c), etc. This
+        is essentially a way to pack a few values together in a record-like
+        structure. In fact, the built-in tuple types are just algebraic product
+        types (and are thus supported in exactly the same way).

 
 

-        The "product" in its name refers to the collection of values belonging
-        to this type. The collection for a product type is the cartesic
+        The \quote{product} in its name refers to the collection of values belonging
+        to this type. The collection for a product type is the Cartesian

         product of the collections for the types of its fields.
 
         product of the collections for the types of its fields.
 

-        These types are translated to \VHDL record types, with one field for
+        These types are translated to \VHDL\ record types, with one field for

         every field in the constructor. This translation applies to all single
         every field in the constructor. This translation applies to all single

-        constructor algebraic datatypes, including those with no fields (unit
-        types) and just one field (which are technically not a product).
+        constructor algebraic datatypes, including those with just one
+        field (which are technically not a product, but generate a VHDL
+        record for implementation simplicity).

       \stopdesc
       \startdesc{Enumerated types}
       \stopdesc
       \startdesc{Enumerated types}

+        \defref{enumerated types}

         An enumerated type is an algebraic datatype with multiple constructors, but
         none of them have fields. This is essentially a way to get an
         enum-like type containing alternatives.
         An enumerated type is an algebraic datatype with multiple constructors, but
         none of them have fields. This is essentially a way to get an
         enum-like type containing alternatives.


@@ -285,7 +485,7 @@ and3 a b c = and (and a b) c
         Note that Haskell's \hs{Bool} type is also defined as an
         enumeration type, but we have a fixed translation for that.
         
         Note that Haskell's \hs{Bool} type is also defined as an
         enumeration type, but we have a fixed translation for that.
         

-        These types are translated to \VHDL enumerations, with one value for
+        These types are translated to \VHDL\ enumerations, with one value for

         each constructor. This allows references to these constructors to be
         translated to the corresponding enumeration value.
       \stopdesc
         each constructor. This allows references to these constructors to be
         translated to the corresponding enumeration value.
       \stopdesc


@@ -293,12 +493,16 @@ and3 a b c = and (and a b) c
         A sum type is an algebraic datatype with multiple constructors, where
         the constructors have one or more fields. Technically, a type with
         more than one field per constructor is a sum of products type, but
         A sum type is an algebraic datatype with multiple constructors, where
         the constructors have one or more fields. Technically, a type with
         more than one field per constructor is a sum of products type, but

-        for our purposes this distinction does not really make a difference,
-        so we'll leave it out.
+        for our purposes this distinction does not really make a
+        difference, so this distinction is note made.

 
 

-        Sum types are currently not supported by the prototype, since its
-        translation is not so obvious as before. The can easily be emulated,
-        as we will see from an example:
+        The \quote{sum} in its name refers again to the collection of values
+        belonging to this type. The collection for a sum type is the
+        union of the the collections for each of the constructors.
+
+        Sum types are currently not supported by the prototype, since there is
+        no obvious \VHDL\ alternative. They can easily be emulated, however, as
+        we will see from an example:

 
         \starthaskell
         data Sum = A Bit Word | B Word 
 
         \starthaskell
         data Sum = A Bit Word | B Word 


@@ -315,55 +519,70 @@ and3 a b c = and (and a b) c
         type Sum = (SumC, Bit, Word, Word)
         \stophaskell
 
         type Sum = (SumC, Bit, Word, Word)
         \stophaskell
 

-        Here, the \hs{SumC} type effectively signals which of the fields of
-        the \hs{Sum} type are valid, all the other ones have no useful value.
+        Here, the \hs{SumC} type effectively signals which of the latter three
+        fields of the \hs{Sum} type are valid (the first two if \hs{A}, the
+        last one if \hs{B}), all the other ones have no useful value.

 
 

-        An obvious problem with this naive approach is the space usage: The
-        example above generates a fairly big \VHDL type. However, we can be
+        An obvious problem with this naive approach is the space usage: the
+        example above generates a fairly big \VHDL\ type. Since we can be

         sure that the two \hs{Word}s in the \hs{Sum} type will never be valid
         sure that the two \hs{Word}s in the \hs{Sum} type will never be valid

-        at the same time, so this is a waste of space.
+        at the same time, this is a waste of space.

 
 

-        Obviously, we could do some duplication detection here to reuse a
+        Obviously, duplication detection could be used to reuse a

         particular field for another constructor, but this would only
         particular field for another constructor, but this would only

-        partially solve the problem. If I would, for example, have an array of
-        8 bits and a 8 bit unsiged word, these are different types and could
-        not be shared. However, in the final hardware, both of these types
-        would simply be 8 bit connections, so we have a 100\% size increase by
-        not sharing these!
+        partially solve the problem. If two fields would be, for
+        example, an array of 8 bits and an 8 bit unsiged word, these are
+        different types and could not be shared. However, in the final
+        hardware, both of these types would simply be 8 bit connections,
+        so we have a 100\% size increase by not sharing these.

       \stopdesc
 
       Another interesting case is that of recursive types. In Haskell, an
       \stopdesc
 
       Another interesting case is that of recursive types. In Haskell, an

-      algebraic datatype can be recursive: Any of its field types can be (or
+      algebraic datatype can be recursive: any of its field types can be (or

       contain) the type being defined. The most well-known recursive type is
       probably the list type, which is defined is:
       
       \starthaskell
       contain) the type being defined. The most well-known recursive type is
       probably the list type, which is defined is:
       
       \starthaskell

-      data List a = Empty | Cons a (List a)
+      data List t = Empty | Cons t (List t)

       \stophaskell
 
       \stophaskell
 

-      Where \hs{Empty} is usually written as \hs{[]} and \hs{Cons} as \hs{:}.
-      This immediately shows the problem with recursive types: What hardware
-      type to allocate here? There is no way to statically determine how long
-      the list will be
+      Note that \hs{Empty} is usually written as \hs{[]} and \hs{Cons} as
+      \hs{:}, but this would make the definition harder to read.  This
+      immediately shows the problem with recursive types: what hardware type
+      to allocate here? 
+      
+      If the naive approach for sum types described above would be used,
+      a record would be created where the first field is an enumeration
+      to distinguish \hs{Empty} from \hs{Cons}. Furthermore, two more
+      fields would be added: one with the (\VHDL\ equivalent of) type
+      \hs{t} (assuming this type is actually known at compile time, this
+      should not be a problem) and a second one with type \hs{List t}.
+      The latter one is of course a problem: this is exactly the type
+      that was to be translated in the first place.

       
       

-      In general, we can say we can never properly translated recursive types:
-      All recursive types have a potentially infinite value (even though in
+      The resulting \VHDL\ type will thus become infinitely deep. In
+      other words, there is no way to statically determine how long
+      (deep) the list will be (it could even be infinite).
+      
+      In general, recursive types can never be properly translated: all
+      recursive types have a potentially infinite value (even though in

       practice they will have a bounded value, there is no way for the
       practice they will have a bounded value, there is no way for the

-      compiler to determine an upper bound on its size.
+      compiler to automatically determine an upper bound on its size).

 
   \subsection{Partial application}
 
   \subsection{Partial application}

-  It should be obvious that we cannot generate hardware signals for all
-  expressions we can express in Haskell. The most obvious criterium for this
-  is the type of an expression. We will see more of this below, but for now it
-  should be obvious that any expression of a function type cannot be
-  represented as a signal or i/o port to a component.
+  Now the translation of application, choice and types has been
+  discussed, a more complex concept can be considered: partial
+  applications. A \emph{partial application} is any application whose
+  (return) type is (again) a function type.

 
 

-  From this, we can see that the above translation rules do not apply to a
-  partial application. \in{Example}[ex:Quadruple] shows an example use of
-  partial application and the corresponding architecture.
+  From this, it should be clear that the translation rules for full
+  application does not apply to a partial application: there are not
+  enough values for all the input ports in the resulting \VHDL.
+  \in{Example}[ex:Quadruple] shows an example use of partial application
+  and the corresponding architecture.

 
 \startbuffer[Quadruple]
 
 \startbuffer[Quadruple]

--- | Multiply the input word by four.
+-- Multiply the input word by four.

 quadruple :: Word -> Word
 quadruple n = mul (mul n)
   where
 quadruple :: Word -> Word
 quadruple n = mul (mul n)
   where


@@ -398,26 +617,29 @@ quadruple n = mul (mul n)
     ncline(mulb)(out);
   \stopuseMPgraphic
 
     ncline(mulb)(out);
   \stopuseMPgraphic
 

-  \placeexample[here][ex:Quadruple]{Simple three port and.}
+  \placeexample[][ex:Quadruple]{Simple three port and.}

     \startcombination[2*1]
       {\typebufferhs{Quadruple}}{Haskell description using function applications.}
       {\boxedgraphic{Quadruple}}{The architecture described by the Haskell description.}
     \stopcombination
 
     \startcombination[2*1]
       {\typebufferhs{Quadruple}}{Haskell description using function applications.}
       {\boxedgraphic{Quadruple}}{The architecture described by the Haskell description.}
     \stopcombination
 

-  Here, the definition of mul is a partial function application: It applies
-  \hs{2 :: Word} to the function \hs{(*) :: Word -> Word -> Word} resulting in
-  the expression \hs{(*) 2 :: Word -> Word}. Since this resulting expression
-  is again a function, we can't generate hardware for it directly. This is
-  because the hardware to generate for \hs{mul} depends completely on where
-  and how it is used. In this example, it is even used twice!
-
-  However, it is clear that the above hardware description actually describes
-  valid hardware. In general, we can see that any partial applied function
-  must eventually become completely applied, at which point we can generate
-  hardware for it using the rules for function application above. It might
-  mean that a partial application is passed around quite a bit (even beyond
-  function boundaries), but eventually, the partial application will become
-  completely applied.
+  Here, the definition of mul is a partial function application: it applies
+  the function \hs{(*) :: Word -> Word -> Word} to the value \hs{2 :: Word},
+  resulting in the expression \hs{(*) 2 :: Word -> Word}. Since this resulting
+  expression is again a function, hardware cannot be generated for it
+  directly.  This is because the hardware to generate for \hs{mul}
+  depends completely on where and how it is used. In this example, it is
+  even used twice.
+
+  However, it is clear that the above hardware description actually
+  describes valid hardware. In general, any partial applied function
+  must eventually become completely applied, at which point hardware for
+  it can be generated using the rules for function application given in
+  \in{section}[sec:description:application]. It might mean that a
+  partial application is passed around quite a bit (even beyond function
+  boundaries), but eventually, the partial application will become
+  completely applied. An example of this principe is given in
+  \in{section}[sec:normalization:defunctionalization].

 
   \section{Costless specialization}
     Each (complete) function application in our description generates a
 
   \section{Costless specialization}
     Each (complete) function application in our description generates a


@@ -425,45 +647,56 @@ quadruple n = mul (mul n)
     design. It is interesting to note that each application of a function
     generates a \emph{separate} piece of hardware. In the final design, none
     of the hardware is shared between applications, even when the applied
     design. It is interesting to note that each application of a function
     generates a \emph{separate} piece of hardware. In the final design, none
     of the hardware is shared between applications, even when the applied

-    function is the same.
+    function is the same (of course, if a particular value, such as the result
+    of a function application, is used twice, it is not calculated twice).

 
 

-    This is distinctly different from normal program compilation: Two separate
+    This is distinctly different from normal program compilation: two separate

     calls to the same function share the same machine code. Having more
     machine code has implications for speed (due to less efficient caching)
     and memory usage. For normal compilation, it is therefore important to
     calls to the same function share the same machine code. Having more
     machine code has implications for speed (due to less efficient caching)
     and memory usage. For normal compilation, it is therefore important to

-    keep the amount of functions limited and maximize the code sharing.
+    keep the amount of functions limited and maximize the code sharing
+    (though there is a tradeoff between speed and memory usage here).

 
     When generating hardware, this is hardly an issue. Having more \quote{code
     sharing} does reduce the amount of \small{VHDL} output (Since different
     component instantiations still share the same component), but after
 
     When generating hardware, this is hardly an issue. Having more \quote{code
     sharing} does reduce the amount of \small{VHDL} output (Since different
     component instantiations still share the same component), but after

-    synthesis, the amount of hardware generated is not affected.
+    synthesis, the amount of hardware generated is not affected. This
+    means there is no tradeoff between speed and memory (or rather,
+    chip area) usage.

 
     In particular, if we would duplicate all functions so that there is a
 
     In particular, if we would duplicate all functions so that there is a

-    duplicate for every application in the program (\eg, each function is then
-    only applied exactly once), there would be no increase in hardware size
-    whatsoever.
+    separate function for every application in the program (\eg, each function
+    is then only applied exactly once), there would be no increase in hardware
+    size whatsoever.
+    
+    Because of this, a common optimization technique called
+    \emph{specialization} can be applied to hardware generation without any
+    performance or area cost (unlike for software). 

    
    

-   TODO: Perhaps these next two sections are a bit too
-   implementation-oriented?
+   \fxnote{Perhaps these next three sections are a bit too
+   implementation-oriented?}

 
     \subsection{Specialization}
 
     \subsection{Specialization}

-      Because of this, a common optimization technique called
-      \emph{specialization} is as good as free for hardware generation.  Given
-      some function that has a \emph{domain} of $D$ (\eg, the set of all
-      possible arguments that could be applied), we create a specialized
-      function with exactly the same behaviour, but with an domain of $D'
-      \subset D$. This subset can be derived in all sort of ways, but commonly
-      this is done by limiting a polymorphic argument to a single type (\eg,
-      removing polymorphism) or by limiting an argument to just a single value
-      (\eg, cross-function constant propagation, effectively removing the
-      argument). 
+      \defref{specialization}
+      Given some function that has a \emph{domain} $D$ (\eg, the set of
+      all possible arguments that the function could be applied to), we
+      create a specialized function with exactly the same behaviour, but
+      with a domain $D' \subset D$. This subset can be chosen in all
+      sorts of ways. Any subset is valid for the general definition of
+      specialization, but in practice only some of them provide useful
+      optimization opportunities.
+
+      Common subsets include limiting a polymorphic argument to a single type
+      (\ie, removing polymorphism) or limiting an argument to just a single
+      value (\ie, cross-function constant propagation, effectively removing
+      the argument). 

       
       Since we limit the argument domain of the specialized function, its
       definition can often be optimized further (since now more types or even
       
       Since we limit the argument domain of the specialized function, its
       definition can often be optimized further (since now more types or even

-      values of arguments are already know). By replacing any application of
+      values of arguments are already known). By replacing any application of

       the function that falls within the reduced domain by an application of
       the specialized version, the code gets faster (but the code also gets
       the function that falls within the reduced domain by an application of
       the specialized version, the code gets faster (but the code also gets

-      bigger, since we now have two versions instead of one!). If we apply
+      bigger, since we now have two versions instead of one). If we apply

       this technique often enough, we can often replace all applications of a
       function by specialized versions, allowing the original function to be
       removed (in some cases, this can even give a net reduction of the code
       this technique often enough, we can often replace all applications of a
       function by specialized versions, allowing the original function to be
       removed (in some cases, this can even give a net reduction of the code


@@ -471,16 +704,16 @@ quadruple n = mul (mul n)
 
       Specialization is useful for our hardware descriptions for functions
       that contain arguments that cannot be translated to hardware directly
 
       Specialization is useful for our hardware descriptions for functions
       that contain arguments that cannot be translated to hardware directly

-      (polymorphic or higher order arguments, for example). If we can create
+      (polymorphic or higher-order arguments, for example). If we can create

       specialized functions that remove the argument, or make it translatable,
       we can use specialization to make the original, untranslatable, function
       obsolete.
 
   \section{Higher order values}
     What holds for partial application, can be easily generalized to any
       specialized functions that remove the argument, or make it translatable,
       we can use specialization to make the original, untranslatable, function
       obsolete.
 
   \section{Higher order values}
     What holds for partial application, can be easily generalized to any

-    higher order expression. This includes partial applications, plain
+    higher-order expression. This includes partial applications, plain

     variables (e.g., a binder referring to a top level function), lambda
     variables (e.g., a binder referring to a top level function), lambda

-    expressions and more complex expressions with a function type (a case
+    expressions and more complex expressions with a function type (a \hs{case}

     expression returning lambda's, for example).
 
     Each of these values cannot be directly represented in hardware (just like
     expression returning lambda's, for example).
 
     Each of these values cannot be directly represented in hardware (just like


@@ -489,39 +722,41 @@ quadruple n = mul (mul n)
     become complete applications, applied lambda expressions disappear by
     applying β-reduction, etc.
 
     become complete applications, applied lambda expressions disappear by
     applying β-reduction, etc.
 

-    So any higher order value will be \quote{pushed down} towards its
+    So any higher-order value will be \quote{pushed down} towards its

     application just like partial applications. Whenever a function boundary
     needs to be crossed, the called function can be specialized.
     application just like partial applications. Whenever a function boundary
     needs to be crossed, the called function can be specialized.

-  
-    TODO: This is section should be improved
+    
+    \fxnote{This section needs improvement and an example}

 
   \section{Polymorphism}
 
   \section{Polymorphism}

-    In Haskell, values can be polymorphic: They can have multiple types. For
+    In Haskell, values can be \emph{polymorphic}: they can have multiple types. For

     example, the function \hs{fst :: (a, b) -> a} is an example of a
     example, the function \hs{fst :: (a, b) -> a} is an example of a

-    polymorphic function: It works for tuples with any element types. Haskell
-    typeclasses allow a function to work on a specific set of types, but the
-    general idea is the same.
+    polymorphic function: it works for tuples with any two element types. Haskell
+    type classes allow a function to work on a specific set of types, but the
+    general idea is the same. The opposite of this is a \emph{monomorphic}
+    value, which has a single, fixed, type.

 
 %    A type class is a collection of types for which some operations are
 %    defined. It is thus possible for a value to be polymorphic while having
 
 %    A type class is a collection of types for which some operations are
 %    defined. It is thus possible for a value to be polymorphic while having

-%    any number of \emph{class constraints}: The value is not defined for
+%    any number of \emph{class constraints}: the value is not defined for

 %    every type, but only for types in the type class. An example of this is
 %    the \hs{even :: (Integral a) => a -> Bool} function, which can map any
 %    value of a type that is member of the \hs{Integral} type class 
 
 %    every type, but only for types in the type class. An example of this is
 %    the \hs{even :: (Integral a) => a -> Bool} function, which can map any
 %    value of a type that is member of the \hs{Integral} type class 
 

-    When generating hardware, polymorphism can't be easily translated. How
-    many wire will you lay down for a value that could have any type? When
+    When generating hardware, polymorphism cannot be easily translated. How
+    many wires will you lay down for a value that could have any type? When

     type classes are involved, what hardware components will you lay down for
     a class method (whose behaviour depends on the type of its arguments)?
     type classes are involved, what hardware components will you lay down for
     a class method (whose behaviour depends on the type of its arguments)?

+    Note that Cλash currently does not allow user-defined type classes,
+    but does partly support some of the built-in type classes (like \hs{Num}).

 
 

-    Fortunately, we can again use the principle of specialization: Since every
-    function application generates separate pieces of hardware, we can know
-    the types of all arguments exactly. Provided that we don't use existential
-    typing, all of the polymorphic types in a function must depend on the
-    types of the arguments (In other words, the only way to introduce a type
-    variable is in a lambda abstraction). Our top level function must not have
-    a polymorphic type (otherwise we wouldn't know the hardware interface to
-    our top level function).
+    Fortunately, we can again use the principle of specialization: since every
+    function application generates a separate piece of hardware, we can know
+    the types of all arguments exactly. Provided that existential typing
+    (which is a \GHC\ extension) is not used typing, all of the
+    polymorphic types in a function must depend on the types of the
+    arguments (In other words, the only way to introduce a type variable
+    is in a lambda abstraction).

 
     If a function is monomorphic, all values inside it are monomorphic as
     well, so any function that is applied within the function can only be
 
     If a function is monomorphic, all values inside it are monomorphic as
     well, so any function that is applied within the function can only be


@@ -529,46 +764,78 @@ quadruple n = mul (mul n)
     specialized to work just for these specific types, removing the
     polymorphism from the applied functions as well.
 
     specialized to work just for these specific types, removing the
     polymorphism from the applied functions as well.
 

+    \defref{entry function}The entry function must not have a
+    polymorphic type (otherwise no hardware interface could be generated
+    for the entry function).
+

     By induction, this means that all functions that are (indirectly) called
     by our top level function (meaning all functions that are translated in
     the final hardware) become monomorphic.
 
   \section{State}
     A very important concept in hardware designs is \emph{state}. In a
     By induction, this means that all functions that are (indirectly) called
     by our top level function (meaning all functions that are translated in
     the final hardware) become monomorphic.
 
   \section{State}
     A very important concept in hardware designs is \emph{state}. In a

-    stateless (or, \emph{combinatoric}) design, every output is a directly and solely dependent on the
+    stateless (or, \emph{combinational}) design, every output is  directly and solely dependent on the

     inputs. In a stateful design, the outputs can depend on the history of
     inputs, or the \emph{state}. State is usually stored in \emph{registers},
     which retain their value during a clockcycle, and are typically updated at
     the start of every clockcycle. Since the updating of the state is tightly
     coupled (synchronized) to the clock signal, these state updates are often
     inputs. In a stateful design, the outputs can depend on the history of
     inputs, or the \emph{state}. State is usually stored in \emph{registers},
     which retain their value during a clockcycle, and are typically updated at
     the start of every clockcycle. Since the updating of the state is tightly
     coupled (synchronized) to the clock signal, these state updates are often

-    called \emph{synchronous}.
+    called \emph{synchronous} behaviour.
+
+    \todo{Sidenote? Registers can contain any (complex) type}

   
   

-    To make our hardware description language useful to describe more that
-    simple combinatoric designs, we'll need to be able to describe state in
-    some way.
+    To make Cλash useful to describe more than simple combinational
+    designs, it needs to be able to describe state in some way.

 
     \subsection{Approaches to state}
       In Haskell, functions are always pure (except when using unsafe
 
     \subsection{Approaches to state}
       In Haskell, functions are always pure (except when using unsafe

-      functions like \hs{unsafePerformIO}, which should be prevented whenever
-      possible). This means that the output of a function solely depends on
-      its inputs. If you evaluate a given function with given inputs, it will
-      always provide the same output.
-
-      TODO: Define pure
-
-      This is a perfect match for a combinatoric circuit, where the output
-      also soley depend on the inputs. However, when state is involved, this
-      no longer holds. Since we're in charge of our own language, we could
-      remove this purity constraint and allow a function to return different
-      values depending on the cycle in which it is evaluated (or rather, the
-      current state). However, this means that all kinds of interesting
-      properties of our functional language get lost, and all kinds of
-      transformations and optimizations might no longer be meaning preserving.
-
-      Provided that we want to keep the function pure, the current state has
-      to be present in the function's arguments in some way. There seem to be
-      two obvious ways to do this: Adding the current state as an argument, or
-      including the full history of each argument.
+      functions with the \hs{IO} monad, which is not supported by Cλash). This
+      means that the output of a function solely depends on its inputs. If you
+      evaluate a given function with given inputs, it will always provide the
+      same output.
+
+      \placeintermezzo{}{
+        \defref{purity}
+        \startframedtext[width=8cm,background=box,frame=no]
+        \startalignment[center]
+          {\tfa Purity}
+        \stopalignment
+        \blank[medium]
+
+        A function is said to be pure if it satisfies two conditions:
+
+        \startitemize[KR]
+        \item When a pure function is called with the same arguments twice, it should
+        return the same value in both cases.
+        \item When a pure function is called, it should have not
+        observable side-effects.
+        \stopitemize
+
+        Purity is an important property in functional languages, since
+        it enables all kinds of mathematical reasoning and
+        optimizattions with pure functions, that are not guaranteed to
+        be correct for impure functions.
+
+        An example of a pure function is the square root function
+        \hs{sqrt}. An example of an impure function is the \hs{today}
+        function that returns the current date (which of course cannot
+        exist like this in Haskell).
+        \stopframedtext
+      }
+
+      This is a perfect match for a combinational circuit, where the output
+      also solely depends on the inputs. However, when state is involved, this
+      no longer holds. Of course this purity constraint cannot just be
+      removed from Haskell. But even when designing a completely new (hardware
+      description) language, it does not seem to be a good idea to
+      remove this purity. This would that all kinds of interesting properties of
+      the functional language get lost, and all kinds of transformations
+      and optimizations are no longer be meaning preserving.
+
+      So our functions must remain pure, meaning the current state has
+      to be present in the function's arguments in some way. There seem
+      to be two obvious ways to do this: adding the current state as an
+      argument, or including the full history of each argument.

 
       \subsubsection{Stream arguments and results}
         Including the entire history of each input (\eg, the value of that
 
       \subsubsection{Stream arguments and results}
         Including the entire history of each input (\eg, the value of that


@@ -580,19 +847,19 @@ quadruple n = mul (mul n)
         An obvious downside of this solution is that on each cycle, all the
         previous cycles must be resimulated to obtain the current state. To do
         this, it might be needed to have a recursive helper function as well,
         An obvious downside of this solution is that on each cycle, all the
         previous cycles must be resimulated to obtain the current state. To do
         this, it might be needed to have a recursive helper function as well,

-        wich might be hard to properly analyze by the compiler.
+        which might be hard to be properly analyzed by the compiler.

 
         A slight variation on this approach is one taken by some of the other
 
         A slight variation on this approach is one taken by some of the other

-        functional \small{HDL}s in the field (TODO: References to Lava,
-        ForSyDe, ...): Make functions operate on complete streams. This means
+        functional \small{HDL}s in the field: \todo{References to Lava,
+        ForSyDe, ...} Make functions operate on complete streams. This means

         that a function is no longer called on every cycle, but just once. It
         takes stream as inputs instead of values, where each stream contains
         all the values for every clockcycle since system start. This is easily
         modeled using an (infinite) list, with one element for each clock
         that a function is no longer called on every cycle, but just once. It
         takes stream as inputs instead of values, where each stream contains
         all the values for every clockcycle since system start. This is easily
         modeled using an (infinite) list, with one element for each clock

-        cycle. Since the funciton is only evaluated once, its output is also a
-        stream. Note that, since we are working with infinite lists and still
-        want to be able to simulate the system cycle-by-cycle, this relies
-        heavily on the lazy semantics of Haskell.
+        cycle. Since the function is only evaluated once, its output must also
+        be a stream. Note that, since we are working with infinite lists and
+        still want to be able to simulate the system cycle-by-cycle, this
+        relies heavily on the lazy semantics of Haskell.

 
         Since our inputs and outputs are streams, all other (intermediate)
         values must be streams. All of our primitive operators (\eg, addition,
 
         Since our inputs and outputs are streams, all other (intermediate)
         values must be streams. All of our primitive operators (\eg, addition,


@@ -600,15 +867,22 @@ quadruple n = mul (mul n)
         well (note that changing a single-element operation to a stream
         operation can done with \hs{map}, \hs{zipwith}, etc.).
 
         well (note that changing a single-element operation to a stream
         operation can done with \hs{map}, \hs{zipwith}, etc.).
 

+        This also means that primitive operations from an existing
+        language such as Haskell cannot be used directly (including some
+        syntax constructs, like \hs{case} expressions and \hs{if}
+        expressions).  This mkes this approach well suited for use in
+        \small{EDSL}s, since those already impose these same
+        limitations. \refdef{EDSL}
+

         Note that the concept of \emph{state} is no more than having some way
         to communicate a value from one cycle to the next. By introducing a
         Note that the concept of \emph{state} is no more than having some way
         to communicate a value from one cycle to the next. By introducing a

-        \hs{delay} function, we can do exactly that: Delay (each value in) a
+        \hs{delay} function, we can do exactly that: delay (each value in) a

         stream so that we can "look into" the past. This \hs{delay} function
         simply outputs a stream where each value is the same as the input
         value, but shifted one cycle. This causes a \quote{gap} at the
         stream so that we can "look into" the past. This \hs{delay} function
         simply outputs a stream where each value is the same as the input
         value, but shifted one cycle. This causes a \quote{gap} at the

-        beginning of the stream: What is the value of the delay output in the
-        first cycle? For this, the \hs{delay} function has a second input
-        (which is a value, not a stream!).
+        beginning of the stream: what is the value of the delay output in the
+        first cycle? For this, the \hs{delay} function has a second input, of
+        which only a single value is used.

 
         \in{Example}[ex:DelayAcc] shows a simple accumulator expressed in this
         style.
 
         \in{Example}[ex:DelayAcc] shows a simple accumulator expressed in this
         style.


@@ -646,7 +920,7 @@ acc in = out
 \stopuseMPgraphic
 
 
 \stopuseMPgraphic
 
 

-        \placeexample[here][ex:DelayAcc]{Simple accumulator architecture.}
+        \placeexample[][ex:DelayAcc]{Simple accumulator architecture.}

           \startcombination[2*1]
             {\typebufferhs{DelayAcc}}{Haskell description using streams.}
             {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
           \startcombination[2*1]
             {\typebufferhs{DelayAcc}}{Haskell description using streams.}
             {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}


@@ -656,14 +930,10 @@ acc in = out
         This notation can be confusing (especially due to the loop in the
         definition of out), but is essentially easy to interpret. There is a
         single call to delay, resulting in a circuit with a single register,
         This notation can be confusing (especially due to the loop in the
         definition of out), but is essentially easy to interpret. There is a
         single call to delay, resulting in a circuit with a single register,

-        whose input is connected to \hs{outl (which is the output of the
-        adder)}, and it's output is the \hs{delay out 0} (which is connected
-        to one of the adder inputs).
+        whose input is connected to \hs{out} (which is the output of the
+        adder), and its output is the expression \hs{delay out 0} (which is
+        connected to one of the adder inputs).

 
 

-        This notation has a number of downsides, amongst which are limited
-        readability and ambiguity in the interpretation. TODO: Reference
-        Christiaan.
-        

       \subsubsection{Explicit state arguments and results}
         A more explicit way to model state, is to simply add an extra argument
         containing the current state value. This allows an output to depend on
       \subsubsection{Explicit state arguments and results}
         A more explicit way to model state, is to simply add an extra argument
         containing the current state value. This allows an output to depend on


@@ -671,18 +941,21 @@ acc in = out
         function pure (letting the result depend only on the arguments), since
         the current state is now an argument.
 
         function pure (letting the result depend only on the arguments), since
         the current state is now an argument.
 

-        In Haskell, this would look like \in{example}[ex:ExplicitAcc].
+        In Haskell, this would look like
+        \in{example}[ex:ExplicitAcc]\footnote[notfinalsyntax]{This
+        example is not in the final Cλash syntax}. \todo{Referencing
+        notfinalsyntax from Introduction.tex doesn't work}

 
 \startbuffer[ExplicitAcc]
 -- input -> current state -> (new state, output)
 acc :: Word -> Word -> (Word, Word)
 
 \startbuffer[ExplicitAcc]
 -- input -> current state -> (new state, output)
 acc :: Word -> Word -> (Word, Word)

-acc in (State s) = (State s', out)
+acc in s = (s', out)

   where
     out = s + in
     s'  = out
 \stopbuffer
 
   where
     out = s + in
     s'  = out
 \stopbuffer
 

-        \placeexample[here][ex:ExplicitAcc]{Simple accumulator architecture.}
+        \placeexample[][ex:ExplicitAcc]{Simple accumulator architecture.}

           \startcombination[2*1]
             {\typebufferhs{ExplicitAcc}}{Haskell description using explicit state arguments.}
             % Picture is identical to the one we had just now.
           \startcombination[2*1]
             {\typebufferhs{ExplicitAcc}}{Haskell description using explicit state arguments.}
             % Picture is identical to the one we had just now.


@@ -693,59 +966,71 @@ acc in (State s) = (State s', out)
         variables are used by a function can be completely determined from its
         type signature (as opposed to the stream approach, where a function
         looks the same from the outside, regardless of what state variables it
         variables are used by a function can be completely determined from its
         type signature (as opposed to the stream approach, where a function
         looks the same from the outside, regardless of what state variables it

-        uses (or wether it's stateful at all).
-
-        This approach is the one chosen for Cλash and will be examined more
-        closely below.
+        uses or whether it is stateful at all).
+        
+        This approach to state has been one of the initial drives behind
+        this research. Unlike a stream based approach it is well suited
+        to completely use existing code and language features (like
+        \hs{if} and \hs{case} expressions) because it operates on normal
+        values. Because of these reasons, this is the approach chosen
+        for Cλash. It will be examined more closely below.

 
     \subsection{Explicit state specification}
 
     \subsection{Explicit state specification}

-      We've seen the concept of explicit state in a simple example below, but
-      what are the implications of this approach?
+      The concept of explicit state has been introduced with some
+      examples above, but what are the implications of this approach?

 
       \subsubsection{Substates}
         Since a function's state is reflected directly in its type signature,
 
       \subsubsection{Substates}
         Since a function's state is reflected directly in its type signature,

-        if a function calls other stateful functions (\eg, has subcircuits) it
+        if a function calls other stateful functions (\eg, has subcircuits), it

         has to somehow know the current state for these called functions. The
         only way to do this, is to put these \emph{substates} inside the
         caller's state. This means that a function's state is the sum of the
         has to somehow know the current state for these called functions. The
         only way to do this, is to put these \emph{substates} inside the
         caller's state. This means that a function's state is the sum of the

-        states of all functions it calls, and its own state.
+        states of all functions it calls, and its own state. This sum
+        can be obtained using something simple like a tuple, or possibly
+        custom algebraic types for clarity.

 
         This also means that the type of a function (at least the "state"
 
         This also means that the type of a function (at least the "state"

-        part) is dependent on its implementation and the functions it calls.
-        This is the major downside of this approach: The separation between
+        part) is dependent on its own implementation and of the functions it
+        calls.
+
+        This is the major downside of this approach: the separation between

         interface and implementation is limited. However, since Cλash is not
         very suitable for separate compilation (see
         \in{section}[sec:prototype:separate]) this is not a big problem in
         interface and implementation is limited. However, since Cλash is not
         very suitable for separate compilation (see
         \in{section}[sec:prototype:separate]) this is not a big problem in

-        practice. Additionally, when using a type synonym for the state type
+        practice.
+
+        Additionally, when using a type synonym for the state type

         of each function, we can still provide explicit type signatures
         while keeping the state specification for a function near its
         definition only.
         of each function, we can still provide explicit type signatures
         while keeping the state specification for a function near its
         definition only.

+        \todo{Example}

     
     

-      \subsubsection{...}
+      \subsubsection{Which arguments and results are stateful?}
+        \fxnote{This section should get some examples}

         We need some way to know which arguments should become input ports and
         which argument(s?) should become the current state (\eg, be bound to
         We need some way to know which arguments should become input ports and
         which argument(s?) should become the current state (\eg, be bound to

-        the register outputs). This does not hold holds not just for the top
-        level function, but also for any subfunctions. Or could we perhaps
+        the register outputs). This does not hold just for the top
+        level function, but also for any subfunction. Or could we perhaps

         deduce the statefulness of subfunctions by analyzing the flow of data
         in the calling functions?
 
         deduce the statefulness of subfunctions by analyzing the flow of data
         in the calling functions?
 

-        To explore this matter, we make an interesting observation: We get
-        completely correct behaviour when we put all state registers in the
-        top level entity (or even outside of it). All of the state arguments
-        and results on subfunctions are treated as normal input and output
-        ports. Effectively, a stateful function results in a stateless
+        To explore this matter, the following observeration is interesting: we
+        get completely correct behaviour when we put all state registers in
+        the top level entity (or even outside of it). All of the state
+        arguments and results on subfunctions are treated as normal input and
+        output ports. Effectively, a stateful function results in a stateless

         hardware component that has one of its input ports connected to the
         output of a register and one of its output ports connected to the
         input of the same register.
 
         hardware component that has one of its input ports connected to the
         output of a register and one of its output ports connected to the
         input of the same register.
 

-        TODO: Example?
+        \todo{Example}

 
         Of course, even though the hardware described like this has the
         correct behaviour, unless the layout tool does smart optimizations,
         there will be a lot of extra wire in the design (since registers will
         not be close to the component that uses them). Also, when working with
         the generated \small{VHDL} code, there will be a lot of extra ports
 
         Of course, even though the hardware described like this has the
         correct behaviour, unless the layout tool does smart optimizations,
         there will be a lot of extra wire in the design (since registers will
         not be close to the component that uses them). Also, when working with
         the generated \small{VHDL} code, there will be a lot of extra ports

-        just to pass one state values, which can get quite confusing.
+        just to pass on state values, which can get quite confusing.

 
         To fix this, we can simply \quote{push} the registers down into the
         subcircuits. When we see a register that is connected directly to a
 
         To fix this, we can simply \quote{push} the registers down into the
         subcircuits. When we see a register that is connected directly to a


@@ -754,7 +1039,7 @@ acc in (State s) = (State s', out)
         trivial when looking at the Haskell code instead of the resulting
         circuit, but the idea is still the same.
 
         trivial when looking at the Haskell code instead of the resulting
         circuit, but the idea is still the same.
 

-        TODO: Example?
+        \todo{Example}

 
         However, when applying this technique, we might push registers down
         too far. When you intend to store a result of a stateless subfunction
 
         However, when applying this technique, we might push registers down
         too far. When you intend to store a result of a stateless subfunction


@@ -764,11 +1049,13 @@ acc in (State s) = (State s', out)
         the called function is in fact stateful. From this we can conclude
         that we have to either:
 
         the called function is in fact stateful. From this we can conclude
         that we have to either:
 

+        \todo{Example of wrong downpushing}
+

         \startitemize
         \item accept that the generated hardware might not be exactly what we
         intended, in some specific cases. In most cases, the hardware will be
         what we intended.
         \startitemize
         \item accept that the generated hardware might not be exactly what we
         intended, in some specific cases. In most cases, the hardware will be
         what we intended.

-        \item explicitely annotate state arguments and results in the input
+        \item explicitly annotate state arguments and results in the input

         description.
         \stopitemize
 
         description.
         \stopitemize
 


@@ -777,10 +1064,9 @@ acc in (State s) = (State s', out)
         end up is easier to implement correctly with explicit annotations, so
         for these reasons we will look at how this annotations could work.
 
         end up is easier to implement correctly with explicit annotations, so
         for these reasons we will look at how this annotations could work.
 

+        \todo{Sidenote: one or more state arguments?}

 
 

-      TODO: Note about conditions on state variables and checking them.
-
-    \subsection{Explicit state annotation}
+    \subsection[sec:description:stateann]{Explicit state annotation}

       To make our stateful descriptions unambigious and easier to translate,
       we need some way for the developer to describe which arguments and
       results are intended to become stateful.
       To make our stateful descriptions unambigious and easier to translate,
       we need some way for the developer to describe which arguments and
       results are intended to become stateful.


@@ -792,11 +1078,11 @@ acc in (State s) = (State s', out)
         result is stateful. This means that the annotation lives
         \quote{outside} of the function, it is completely invisible when
         looking at the function body.
         result is stateful. This means that the annotation lives
         \quote{outside} of the function, it is completely invisible when
         looking at the function body.

-        \item Use some kind of annotation on the type level, \eg give stateful
-        arguments and (part of) results a different type. This has the
+        \item Use some kind of annotation on the type level, \ie\ give stateful
+        arguments and stateful (parts of) results a different type. This has the

         potential to make this annotation visible inside the function as well,
         such that when looking at a value inside the function body you can
         potential to make this annotation visible inside the function as well,
         such that when looking at a value inside the function body you can

-        tell if it's stateful by looking at its type. This could possibly make
+        tell if it is stateful by looking at its type. This could possibly make

         the translation process a lot easier, since less analysis of the
         program flow might be required.
       \stopitemize
         the translation process a lot easier, since less analysis of the
         program flow might be required.
       \stopitemize


@@ -805,19 +1091,16 @@ acc in (State s) = (State s', out)
       implemented in Cλash. \in{Section}[sec:prototype:statetype] expands on
       the possible ways this could have been implemented.
 
       implemented in Cλash. \in{Section}[sec:prototype:statetype] expands on
       the possible ways this could have been implemented.
 

-  TODO: Say something about dependent types and fixed size vectors
+  \todo{Note about conditions on state variables and checking them}

 
   \section[sec:recursion]{Recursion}
 
   \section[sec:recursion]{Recursion}

-  An import concept in functional languages is recursion. In it's most basic
-  form, recursion is a function that is defined in terms of itself. This
+  An important concept in functional languages is recursion. In its most basic
+  form, recursion is a definition that is described in terms of itself. A
+  recursive function is thus a function that uses itself in its body. This

   usually requires multiple evaluations of this function, with changing
   arguments, until eventually an evaluation of the function no longer requires
   itself.
 
   usually requires multiple evaluations of this function, with changing
   arguments, until eventually an evaluation of the function no longer requires
   itself.
 

-  Recursion in a hardware description is a bit of a funny thing. Usually,
-  recursion is associated with a lot of nondeterminism, stack overflows, but
-  also flexibility and expressive power.
-

   Given the notion that each function application will translate to a
   component instantiation, we are presented with a problem. A recursive
   function would translate to a component that contains itself. Or, more
   Given the notion that each function application will translate to a
   component instantiation, we are presented with a problem. A recursive
   function would translate to a component that contains itself. Or, more


@@ -826,18 +1109,35 @@ acc in (State s) = (State s', out)
   problem for generating hardware.
 
   This is expected for functions that describe infinite recursion. In that
   problem for generating hardware.
 
   This is expected for functions that describe infinite recursion. In that

-  case, we can't generate hardware that shows correct behaviour in a single
+  case, we cannot generate hardware that shows correct behaviour in a single

   cycle (at best, we could generate hardware that needs an infinite number of
   cycles to complete).
   
   cycle (at best, we could generate hardware that needs an infinite number of
   cycles to complete).
   

-  However, most recursive hardware descriptions will describe finite
+  \placeintermezzo{}{
+    \startframedtext[width=8cm,background=box,frame=no]
+    \startalignment[center]
+      {\tfa \hs{null}, \hs{head} and \hs{tail}}
+    \stopalignment
+    \blank[medium]
+      The functions \hs{null}, \hs{head} and \hs{tail} are common list
+      functions in Haskell. The \hs{null} function simply checks if a list is
+      empty. The \hs{head} function returns the first element of a list. The
+      \hs{tail} function returns containing everything \emph{except} the first
+      element of a list.
+
+      In Cλash, there are vector versions of these functions, which do exactly
+      the same.
+    \stopframedtext
+  }
+
+  However, most recursive definitions will describe finite

   recursion. This is because the recursive call is done conditionally. There
   recursion. This is because the recursive call is done conditionally. There

-  is usually a case statement where at least one alternative does not contain
+  is usually a \hs{case} expression where at least one alternative does not contain

   the recursive call, which we call the "base case". If, for each call to the
   the recursive call, which we call the "base case". If, for each call to the

-  recursive function, we would be able to detect which alternative applies,
-  we would be able to remove the case expression and leave only the base case
-  when it applies. This will ensure that expanding the recursive functions
-  will terminate after a bounded number of expansions.
+  recursive function, we would be able to detect at compile time which
+  alternative applies, we would be able to remove the \hs{case} expression and
+  leave only the base case when it applies. This will ensure that expanding
+  the recursive functions will terminate after a bounded number of expansions.

 
   This does imply the extra requirement that the base case is detectable at
   compile time. In particular, this means that the decision between the base
 
   This does imply the extra requirement that the base case is detectable at
   compile time. In particular, this means that the decision between the base


@@ -848,8 +1148,9 @@ acc in (State s) = (State s', out)
   passed in as an argument. Since we represent lists as vectors that encode
   the length in the vector type, it seems easy to determine the base case. We
   can simply look at the argument type for this. However, it turns out that
   passed in as an argument. Since we represent lists as vectors that encode
   the length in the vector type, it seems easy to determine the base case. We
   can simply look at the argument type for this. However, it turns out that

-  this is rather non-trivial to write down in Haskell in the first place. As
-  an example, we would like to write down something like this:
+  this is rather non-trivial to write down in Haskell already, not even
+  looking at translation. As an example, we would like to write down something
+  like this:

  
   \starthaskell
     sum :: Vector n Word -> Word
  
   \starthaskell
     sum :: Vector n Word -> Word


@@ -858,53 +1159,65 @@ acc in (State s) = (State s', out)
       False -> head xs + sum (tail xs)
   \stophaskell
 
       False -> head xs + sum (tail xs)
   \stophaskell
 

-  However, the typechecker will now use the following reasoning (element type
-  of the vector is left out):
+  However, the Haskell typechecker will now use the following reasoning.
+  For simplicity, the element type of a vector is left out, all vectors
+  are assumed to have the same element type. Below, we write conditions
+  on type variables before the \hs{=>} operator. This is not completely
+  valid Haskell syntax, but serves to illustrate the typechecker
+  reasoning. Also note that a vector can never have a negative length,
+  so \hs{Vector n} implicitly means \hs{(n >= 0) => Vector n}.

 
 

+  \todo{This typechecker disregards the type signature}

   \startitemize
   \item tail has the type \hs{(n > 0) => Vector n -> Vector (n - 1)}
   \item This means that xs must have the type \hs{(n > 0) => Vector n}
   \item This means that sum must have the type \hs{(n > 0) => Vector n -> a}
   \startitemize
   \item tail has the type \hs{(n > 0) => Vector n -> Vector (n - 1)}
   \item This means that xs must have the type \hs{(n > 0) => Vector n}
   \item This means that sum must have the type \hs{(n > 0) => Vector n -> a}

+  (The type \hs{a} is can be anything at this stage, we will not try to finds
+  its actual type in this example).

   \item sum is called with the result of tail as an argument, which has the
   \item sum is called with the result of tail as an argument, which has the

-  type \hs{Vector n} (since \hs{(n > 0) => n - 1 == m}).
+  type \hs{Vector n} (since \hs{(n > 0) => Vector (n - 1)} is the same as \hs{(n >= 0)
+  => Vector n}, which is the same as just \hs{Vector n}).

   \item This means that sum must have the type \hs{Vector n -> a}
   \item This is a contradiction between the type deduced from the body of sum
   (the input vector must be non-empty) and the use of sum (the input vector
   could have any length).
   \stopitemize
 
   \item This means that sum must have the type \hs{Vector n -> a}
   \item This is a contradiction between the type deduced from the body of sum
   (the input vector must be non-empty) and the use of sum (the input vector
   could have any length).
   \stopitemize
 

-  As you can see, using a simple case at value level causes the type checker
-  to always typecheck both alternatives, which can't be done! This is a
-  fundamental problem, that would seem perfectly suited for a type class.
-  Considering that we need to switch between to implementations of the sum
-  function, based on the type of the argument, this sounds like the perfect
-  problem to solve with a type class. However, this approach has its own
-  problems (not the least of them that you need to define a new typeclass for
-  every recursive function you want to define).
-
-  Another approach tried involved using GADTs to be able to do pattern
-  matching on empty / non empty lists. While this worked partially, it also
-  created problems with more complex expressions.
-
-  TODO: How much detail should there be here? I can probably refer to
-  Christiaan instead.
-
-  Evaluating all possible (and non-possible) ways to add recursion to our
-  descriptions, it seems better to leave out list recursion alltogether. This
-  allows us to focus on other interesting areas instead. By including
-  (builtin) support for a number of higher order functions like map and fold,
-  we can still express most of the things we would use list recursion for.
- 
+  As you can see, using a simple \hs{case} expression  at value level causes
+  the type checker to always typecheck both alternatives, which cannot be
+  done. The typechecker is unable to distinguish the two case
+  alternatives (this is partly possible using \small{GADT}s, but that
+  approach faced other problems \todo{ref christiaan?}).
+
+  This is a fundamental problem, that would seem perfectly suited for a
+  type class.  Considering that we need to switch between to
+  implementations of the sum function, based on the type of the
+  argument, this sounds like the perfect problem to solve with a type
+  class. However, this approach has its own problems (not the least of
+  them that you need to define a new type class for every recursive
+  function you want to define).
+
+  \todo{This should reference Christiaan}
+

   \subsection{General recursion}
   Of course there are other forms of recursion, that do not depend on the
   length (and thus type) of a list. For example, simple recursion using a
   counter could be expressed, but only translated to hardware for a fixed
   number of iterations. Also, this would require extensive support for compile
   time simplification (constant propagation) and compile time evaluation
   \subsection{General recursion}
   Of course there are other forms of recursion, that do not depend on the
   length (and thus type) of a list. For example, simple recursion using a
   counter could be expressed, but only translated to hardware for a fixed
   number of iterations. Also, this would require extensive support for compile
   time simplification (constant propagation) and compile time evaluation

-  (evaluation constant comparisons), to ensure non-termination. Even then, it
-  is hard to really guarantee termination, since the user (or GHC desugarer)
-  might use some obscure notation that results in a corner case of the
-  simplifier that is not caught and thus non-termination.
+  (evaluation of constant comparisons), to ensure non-termination.
+  Supporting general recursion will probably require strict conditions
+  on the input descriptions. Even then, it will be hard (if not
+  impossible) to really guarantee termination, since the user (or \GHC\
+  desugarer) might use some obscure notation that results in a corner
+  case of the simplifier that is not caught and thus non-termination.
+
+  Evaluating all possible (and non-possible) ways to add recursion to
+  our descriptions, it seems better to limit the scope of this research
+  to exclude recursion. This allows for focusing on other interesting
+  areas instead. By including (built-in) support for a number of
+  higher-order functions like \hs{map} and \hs{fold}, we can still
+  express most of the things we would use (list) recursion for.
+  

 
 

-  Due to these complications, we leave other forms of recursion as
-  future work as well.
+% vim: set sw=2 sts=2 expandtab: